Warp effects in EFT Formulating the problem Hamiltonian approach Application to warped compactifications Kähler metric in the deformed conifold Kinetic terms in warped compactifications Gonzalo Torroba Department of Physics, NHETC, Rutgers University String Phenomenology 2008, UPenn Based on [M. Douglas, GT; arXiv:0805.3700] and work in collaboration with G. Shiu, B. Underwood, J. Shelton Abstract : We develop formalism for computing kinetic terms in string compactifications with warping. This is based on the Hamiltonian of GR. Physical fluctuations turn out to obey a harmonic-type gauge condition, but depending on the warp factor. As an application, we work out the kinetic term of the complex modulus in the warped deformed conifold. Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem Hamiltonian approach Application to warped compactifications Kähler metric in the deformed conifold The aim is to understand better the sugra limit of string theory. Requiring 4d maximal symmetry (AdS, Mink, dS), the most general background is a warped product, ds 2 = e 2 A ( y ; u ) g µν ( x ) dx µ dx ν + g ij ( y ; u ) dy i dy j where e 2 A is the warp factor, and the internal metric g ij depends on some parameters u I . How are warp effects encoded in the low energy dynamics? This turns out to be very hard to understand and there is still a lot of work in progress in this direction. Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem N = 1 case Hamiltonian approach Example: deformed conifold Application to warped compactifications However... Kähler metric in the deformed conifold Warp effects in EFT The warp factor arises from backreaction of branes/fluxes on geometry. Some examples: - AdS/CFT - exponential hierarchies and low scale susy breaking - dualities with confining gauge theories. (Kähler potential?) An important question common to all these examples is what is the 4d EFT description for the previous metric fluctuations u I . It turns out that the effects of warping are encoded mainly in the kinetic terms. To see this, we will consider an example. Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem N = 1 case Hamiltonian approach Example: deformed conifold Application to warped compactifications However... Kähler metric in the deformed conifold N = 1 case Consider type IIb with BPS fluxes and branes [DRS; GKP] . This preserves N = 1 in 4d. The internal manifold is conformally equivalent to a CY, with the conformal and warp factors related: 10 = e 2 A ( y ; u ) η µν dx µ dx ν + e − 2 A ( y ; u ) ˜ ds 2 g ij ( y ; u ) dy i dy j g ij is the CY metric and { u I } represents both complex and Kähler moduli. ˜ Then the effective field theory for { u I } is described by the usual sugra expression J D I W D ¯ J W ∗ − 3 | W | 2 � G I ¯ V = e K � Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem N = 1 case Hamiltonian approach Example: deformed conifold Application to warped compactifications However... Kähler metric in the deformed conifold - W = W GVW is not affected by warping - so, do warp effects come from e K or G I ¯ J ? � Conjecture by [DeWolfe-Giddings] : warp corrections in sugra given by � � β = − 1 � e − 4 A Ω ∧ ¯ � e − 4 A χ α ∧ ¯ K = − log Ω ⇒ G α ¯ χ β V W This is suggested by the fact that � � � � d 6 y ˜ d 6 y ˜ g 6 e − 4 A ( y ) V CY = g 6 → V W = To understand better this proposal, let’s look at the warped deformed conifold [Klebanov-Strassler] Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem N = 1 case Hamiltonian approach Example: deformed conifold Application to warped compactifications However... Kähler metric in the deformed conifold Example: deformed conifold The complex modulus S parametrizes the size of the deformed 3-cycle, through which there are N units of F 3 flux. [Douglas, Shelton, GT] computed the warp corrections to the Kähler metric: c log Λ 3 | S | + c ′ ( g s N α ′ ) 2 1 � � 0 S = − ∂ S ∂ ¯ G S ¯ S K = V W | S | 4 / 3 the log piece is the (unwarped) N = 2 contribution the warp factor introduces a new type of | S | − 4 / 3 divergence Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem N = 1 case Hamiltonian approach Example: deformed conifold Application to warped compactifications However... Kähler metric in the deformed conifold Near the conifold point the new term dominates, producing large changes in the EFT: V ∝ | S | 4 / 3 | D S W | 2 Furthermore, in a model that breaks susy at small enough S , G S ¯ S will produce a parametrically small scale of susy breaking. V 0.08 0.06 0.04 0.02 S 0.5 1 1.5 2 Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem N = 1 case Hamiltonian approach Example: deformed conifold Application to warped compactifications However... Kähler metric in the deformed conifold However... This proposal suffers from some problems, kinematics: the conjectured Kähler metric β = − 1 � e − 4 A χ α ∧ ¯ G α ¯ χ β V W is not diff invariant ( χ → χ + d λ ). See also [Giddings and Maharana; Douglas, Shiu, GT, Underwood] dynamics: new light KK modes ... [Douglas, Shiu, GT, Underwood] In any case, the upshot is that, quite generally, one expects kinetic terms to contain the main effect of warping. Therefore we need to develop a method for computing kinetic terms in general warped backgrounds. Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem Hamiltonian approach Compensators in Yang-Mills theory Application to warped compactifications Kähler metric in the deformed conifold Formulating the problem Start from a general warped solution which depends on certain parameters u I , ds 2 = e 2 A ( y ; u ) g µν ( x ) dx µ dx ν + g ij ( y ; u ) dy i dy j The standard procedure to compute 4d kinetic terms is to promote u I → u I ( x ) and extract d 4 x √ g 4 g µν G IJ ( u ) ∂ µ u I ∂ ν u I � � R 10 → Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem Hamiltonian approach Compensators in Yang-Mills theory Application to warped compactifications Kähler metric in the deformed conifold Formulating the problem Start from a general warped solution which depends on certain parameters u I , ds 2 = e 2 A ( y ; u ) g µν ( x ) dx µ dx ν + g ij ( y ; u ) dy i dy j The standard procedure to compute 4d kinetic terms is to promote u I → u I ( x ) and extract d 4 x √ g 4 g µν G IJ ( u ) ∂ µ u I ∂ ν u I � � R 10 → [Giddings, Maharana] emphasized that this is not consistent, because � y ; u ( x ) � g MN doesn’t solve the 10d eoms. This turns out to be equivalent to the failure of G IJ to be diff invariant. Extra terms (proportional to derivatives ∂ u . . . ) are needed, to compensate for the time-dependence of u ( x ) . Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem Hamiltonian approach Compensators in Yang-Mills theory Application to warped compactifications Kähler metric in the deformed conifold Compensators in Yang-Mills theory To understand the effect of compensating fields, consider a U ( 1 ) gauge field d 10 x √ g 10 F MN F MN S = − 1 � 4 and a family of solutions to D i F ij = 0 parametrized by u I , � � A M = A µ = 0 , A i ( y ; u ) Substituting u I → u I ( x ) , the kinetic terms give the metric d 6 y √ g 6 g ij ∂ A i � ∂ A j G IJ = ∂ u I ∂ u J However, this expression is not invariant under δ A i = ∂ i ǫ . Since the original 10d action is invariant, there should be an error in the dimensional reduction. Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem Hamiltonian approach Compensators in Yang-Mills theory Application to warped compactifications Kähler metric in the deformed conifold The error is in assuming that A µ = 0 still holds for time-dependent moduli: D M F M µ = 0 ⇒ ∂ µ ∂ i A i = ∂ i ∂ i A µ cannot be solved by ∂ µ A i � = 0 , A µ = 0 The new time-dependence forces a nonzero 4d component A µ = Ω I ∂ µ u I , ∂ i ∂ i Ω I = ∂ i ∂ A i ∂ u I This is the simplest example of a compensating field . The only effect of the compensator is to shift ∂ A i ∂ u I → δ I A i := ∂ A i ∂ u I − ∂ i Ω I so that ∂ i ( δ I A i ) = 0 The field space metric is simply d 6 y √ g 6 g ij δ I A i δ J A j � G IJ = Gonzalo Torroba Kinetic terms in warped compactifications
Warp effects in EFT Formulating the problem Hamiltonian of GR Hamiltonian approach Kinetic terms Application to warped compactifications Kähler metric in the deformed conifold Hamiltonian approach In analogy with the YM case, in warped compactifications time-dependent parameters will source off-diagonal components of the metric: 10 = e 2 A ( y ; u ) g µν ( x ) dx µ dx ν + B ( I ) j ( y ) ∂ µ u I dx µ dy j + g ij ( y ; u ) dy i dy j ds 2 However, the YM approach is hard to generalize to this case... √ It turns out that a direct way for finding the right gauge invariant kinetic terms is to derive the Hamiltonian of such warped backgrounds. Gonzalo Torroba Kinetic terms in warped compactifications
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